On Nov 9, 2007, at 12:26 PM, Vaughan Pratt wrote:
> ... whereas the definition in your proof allowed manual
> restarting. The DARPA Grand Challenge race disallowed any team
> assistance during the race, which seems equally applicable here.
>> ...
>> I see two challenges here. First, can you replace outside
> intervention
> by a single infinite initial condition that encodes at one time the
> infinitely many restarts your proof was performing manually, so that
> the
> subject 2,3 machine can do its own restarting?
My understanding is that this is in fact the way Smith's construction
works. There are no restarts, merely a single run from an infinite
initial condition. Did I misunderstand?
To me, these are the issues, in decreasing order of importance:
1. What is the nature of the computation that produces the infinite
initial condition?
2. If this computation is agreed to be "sufficiently simple", is it
justifiable to provide an infinite non-repeating pattern to a
universal Turing Machine? (I don't believe it's even universally
agreed yet that it's justifiable to provide an infinite repeating
input. The case of Wolfram's "rule 110", which does use an infinite
repeating input, is different, because it is a CA, where all cells
recompute at the same time; it makes some sense to give each cell a
finite initial input.)
3. If we agree that the above are not problems, and the (2, 3) machine
should be called universal, is it meaningful to refer to it, as
Wolfram does, as "the smallest universal Turing machine that exists"?
There are many kinds of computation machine. It seems to me that to
compete in a game of minimizing states * symbols, for a result to be
meaningful there must be an established definition of what kind of
machine one is using. In this case a lot of what would normally be
done by the Turing Machine is offloaded to pre- and post-processing,
which is a way of redefining the problem to suit your (Wolfram's)
purposes. How can we compare the information content of this kind of
Turing Machine to a "standard" universal Turing Machine? And how can
we compare it to, say, Conway's Game of Life, or Post's tag systems,
or any other deterministic models of computation, let alone
nondeterministic models? (There is a game model of computation that is
undecidable using finite resources, e.g.)
There is also a more troubling meta-issue with this proof, which is
the way Wolfram Research has handled it. As previously mentioned, the
prize committee evidently did not sign off on the proof; Wolfram
Research did. FOM postings from Wolfram Research have raised a number
of excellent points, correctly in my view pointing out that issues
such as the above are worthwhile, and that discussion of them is a
contribution to the literature. However, that does not seem to be what
is happening. There is no mention of the implicit redefinition of
universality on www.wolframscience.com. Instead, we see "The lower
limit on Turing machine universality is proved -- providing new
evidence for Wolfram's Principle of Computational Equivalence."
Evidently Smith's proof will be published in the journal Complex
Systems, so one could hope for objective peer review. However, this
journal is an arm of Wolfram Research.
This entire episode seems to me a massive perversion of the scientific
process.
Incidentally, I am giving a short talk on this topic this evening, so
if anyone has any immediate feedback to the above I would appreciate
hearing it.
Bob Hearn
---------------------------------------------
Robert A. Hearn
Neukom Institute for Computational Science, Dartmouth College
robert.a.hearn at dartmouth.eduhttp://www.dartmouth.edu/~rah/